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The Nodal Point Determining the location of the Nodal Point Some examples using the Laser Pointer method, including the Sigma 8mm Fisheye Lens The Nodal Point of the lens can be considered as the point at which the rays entering the lens converge. It can also be considered as the centre of perspective of the lens or the apparent pupil. This point is actually the Front Nodal Point as the lens also has a Rear Nodal Point and in a simple lens the two nodes converge to a single point. It is important to know the location of the Nodal Point for Photogrammetric purposes. To enable the correct origin and orientation in space of the rays used to compute the intersection when using the technique of Photographic Intersection it is important that the Front Nodal Point of the lens is determined as accurately as possible. It is also vital that the Front Nodal Point of the lens is known for applications such as using the photographic image on conjunction with HDS (High Definition Surveying) data and for any application for where an equirectangular is to be produced using software such as PTGui.
Photogrammetric camera lenses are constructed so as to be 'symmetrical'. The mounting of the Wild P32, shown in the image, on a theodolite was arranged so that the front nodal point of the lens was coincident with the standing axis of the theodolite when the telescope was truly horizontal even though this meant that the camera was not well balanced. In normal photographic lenses, such as those we would select for Photographic Intersection, the front nodal point is often not a single point but 'slides' along the principal ray of the lens depending on the angle between the ray being considered and the principal ray. For the purposes of Photographic Intersection the position of the front nodal point should be determined for the widest angle of view for the lens, although there is the opportunity to introduce the different locations of the Nodal Point as another parameter in the computation process. Determining the location of the Nodal Point The Nodal Point can be measured as part of the camera and lens calibration process using a calibration rig constructed for determining the parameters of the camera and lens for photogrammetric purposes. Such setups are few and far between, and even if you do locate one, the cost of the calibration will probably be more than the value of the equipment you are calibrating, which defeats the object of using Photographic Intersection as a low cost method of accurately measuring 3D points. It is therefore desirable to find a way of determining the Nodal Point yourself with easy to obtain and low cost items. At the end of the 1970s, beginning of the 1980s, I used a method similar to that described by Michel Thoby with nails and banknotes except that I used needles and the face of an "E" survey staff or tape measure.
The technique is to make an image of the scene, then use the detail of the background object obscured by the foreground objects to reconstruct the rays that converge at the Nodal Point and thus define it. It is imperative that nothing in the arrangement must be disturbed whilst a suitable print of the scene is made so that the rays can be accurately drawn. In the digital age this is not so much of a problem but when film is used keeping the scene intact can be a problem. A disadvantage of this method is that using nails or needles with a background quite near to the lens is that the lens is close focused to keep the detail sharp in the image, but that the lens will more likely to be focused at or near infinity when actually used for measurement. The Nodal Point can change with focusing in some lenses. This can be overcome by going outside and using canes, ranging rods or dowels as the foreground objects with say a building facade as the background image and with the camera and lens on a "Plane Table". Keeping the arrangement undisturbed whilst processing film and printing a decent size image was even more of a challenge! Note that the camera has been rotated around its principal ray so that the diagonal of the field of view is used to get the maximum angle of view, a technique that can be used with both the other methods following.
A simpler method of determining the Nodal Point is to place an object (thin pole or wire) near the lens and align it with a more distant object. If the camera is rotated about the Front Nodal Point the two objects will be aligned when they are in the centre of the image and also when at each side of the image. If the objects are aligned when in the centre of the image but appear to separate as the camera is rotated then the rotation is not about the Nodal Point. This method may be simpler than the first method described, but is more fiddly and less accurate.
This method is remarkably precise and accurate results can be obtained quickly and easily. The position of the lens, especially the front which will be used as the reference, can be orthographically projected onto the paper using a simple set square.
Results from using the Laser Pointer method to determine the location of the Nodal Point The position of the Nodal Point, and its behaviour, clearly depends on the lens design and construction.
It would be interesting to see how other lenses behave, especially lenses such as the Sigma 4.5mm Fisheye, Sigma 10mm and 15mm Diagonal Fisheye and Nikon 10.5mm Fisheye. I would be happy to add these lenses as examples here if anyone is prepared to loan me the lens to work with.
An application where the location of the Nodal Point is particularly important is in panoramic photography. As discussed in Creating Your Own Panoramas, the location of the Nodal Point is of little interest when taking images for a panorama of scenery which is some distance from the camera, but if the subject is close to the camera or the images are to be used for creating full 360° panorama, then the camera should be rotated about the Nodal Point.
The first consideration is that the camera should be positioned so that the principal ray of the lens coincides with the vertical axis of rotation of the camera. As the Nodal Point(s) lie along the principal ray it (they) will be in the same plane as the vertical axis of rotation, but the camera should then be positioned so that the Nodal Point coincides with this axis. With a lens such as the Sigma 10-20mm this is easily achieved using a Panoramic Head with the single Nodal Point located at the centre of rotation. Panoramic Heads are available from a number of sources such as Panosauras, Ninja and Manfrotto and range in price and sophistication, the more expensive ones having user settable click stop to accurately position the camera for each exposure. When I first started to explore the world of 360° panoramas I built my own from scraps I had from previous experiments. The advantage was that I had a very light and easily portable panoramic had for very little cost, but the disadvantage was that the head was built around my Nikon D70 and could not be easily modified to accommodate other cameras, so I decided to invest in a Panosaurus. Positioning the Sigma 10-20mm was straight forward, but the disadvantage of using a rectilinear or ‘normal’ lens is the number of exposures required to achieve a good overlap for the software to create the equirectangular image from. As the table following shows, the number of images required rapidly increases as the focal length of the lens increases.
I usually also take an additional image to the zenith (vertically upwards) which is sometimes useful in the creation of the panorama, especially in situations such as outdoors where clouds in the sky will be in slightly different places for each image.
To reduce the
number of images a ‘fisheye’ lens such as the Sigma 8mm can be used,
but if we try and visualise the ‘shape’ the rays make inside the
lens we can see that the perfect cone of a lens such as the Sigma
10-20mm is very different to the complex geometry
With a fisheye lens, such as the Sigma 8mm and a full size 35mm format, the image is a full circle so only three images at 120° intervals are required, plus one to the zenith. With the same lens and an APS-C sensor either six images at 60° intervals or eight images at 45° intervals gives good results It is possible the ‘get away’ with four images at 90° intervals, but it can be very difficult to get a good result if the overlaps have little or no detail. Taking too many images, such as 12 at 30°, results in too many of images having common areas, which can result in a poor solution using software such as PTGui, or even no solution, but in such a case every other image can be excluded and two solution, each of six images, can be achieved. Getting to the site and setting up the camera is the most time consuming part of the exercise so I tend to take two sets, one of six images and one of eight, especially when shooting digitally where there is no consideration for the cost of film. If the tripod in the panoramic image is to be replaced with a "tripod cap" image the camera can be tilted upwards by 10° to 15° so that a zenith image is not necessary, but it is usually worth taking one anyway, especially outdoors where clouds in the sky will be in slightly different places for each image. An important consideration is that if the fisheye lens does produce a full circle on the sensor, such as with the Sigma 8mm and a full 35mm format, that only some 54% if the image is used (i.e. with a 10Mp sensor only 5.4Mp are used) whereas 92% of the image is used with the same lens and an APS-C sensor (i.e. 9.2Mp with a 10Mp sensor ).
If you are interested in Photographic Intersection, or wish to comment on the subject, please contact me on PhotoIntersect@AOL.com.
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The Nodal Point is also the ideal point to rotate your lens around
when taking images to 'stitch' together to produce panoramic images.
If you are 
The most innovative
and inspired method I have come across recently is that used by
Using
the knowledge of the location of the Nodal Point of a lens will
depend on the application. For a Photogrammetric application, such
as Photographic Intersection, the displacement (offsets) of the
Nodal Point from the centre of rotation of the Instrument on which
the camera is mounted is used to derive the point of origin of a ray
in space passing through a point digitised in the image. In the
case of a lens, such as the Sigma 10-20mm where the Nodal Point is a
single point for a given focal length, the Nodal Point will describe
a sphere around the centre of rotation of the Instrument and the
displacement is easily applied in the mathematics. If a lens, such
as the Nikon 28mm is used, where the Nodal Point moves along the
principal ray depending on the angle of view, then the mathematics
is a little more complex. However the first part of the calculation
for the point digitised in the image can be used to calculate the
angle the ray makes with the principal ray so the correct location
of the Nodal Point can be determined.
in
a lens such as the Sigma 8mm, which is in effect lens distortion.
We can position the Sigma 10-20mm so that the Nodal Point is
actually at the centre of rotation of the panoramic head, but where
should we position the Sigma 8mm? Should we position it so that the
gold ring is coincident with the vertical axis of rotation, which is
where the rays close to the principal ray have their Nodal Points,
or should we position it so that that the front of the lens is
coincident with the vertical axis of rotation, which is where the
rays at the widest angle of view have their Nodal Points, or
somewhere in between? I decided to investigate this empirically and
found that I could not determine any difference in the results so
have settle for “somewhere in between”, but far enough forward to
avoid the panoramic head encroaching into the side of the image.
There are
applications where it is important that the Nodal Point coincides
with the location of the centre of rotation of another device. HDS
(High Definition Surveying) scanners produce a ‘point cloud’ where
each point in the image has a 3D co-ordinate, but the colour of the
point is determined by the reflectivity of the surface that the
point is on. This information is useful in itself, but many users
would like to have the ‘true’ colours of the points in the point
cloud and this can be achieved by replacing the scanner with the
camera on a panoramic head where the Nodal Point of the lens is
coincident with the centre of rotation of the scanner and creating a
full 360°panorama which can then be related to the scanned points.
